True cofinality tcf

Question

Please, here on the page 6, in the definition 0.11 I wonder what happens with the definition of "tcf" when $I$ is this poset:

o     o
 \   /
  \ /
   o

This $I$ has no $<_*-$ increasing sequence, am I right?

Answer 1

Yes, you are correct. More or less.

Note that the true cofinality is always a regular cardinal and never larger than the size of the partial order.

Since arguably $0$ is excluded by the requirement of a cofinal sequence, that means that the true cofinality of a finite partial order is either $1$ if there is a maximum, or "undefined" (which might as well be $0$) otherwise.

Of course, the definition is meant to be applied to partial orders given by $\prod_{i\in I}\lambda_i/F$ where $I$ is an infinite set of infinite cardinals, usually regular, and $F$ is a filter/ideal on $I$. From that perspective finite objects have no bearing on anything, but they do make good toys on occasion.